Symmetric Willmore surfaces of revolution satisfying natural
boundary conditions
Matthias Bergner, Anna Dall’Acqua∗, Steffen Fröhlich
April 29, 2009
Abstract
We consider the Willmore-type functional
Z
Z
2
Wγ (Γ) :=
H dA − γ K dA,
Γ
Γ
where H and K denote mean and Gaussian curvature of a surface Γ, and γ ∈ [0, 1] is a real
parameter. Using direct methods of the calculus of variations, we prove existence of surfaces of
revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation
corresponding to Wγ and which satisfy the following boundary conditions: the height at the
boundary is prescribed, and the second boundary condition is the natural one when considering
critical points where only the position at the boundary is fixed. In the particular case γ = 0
the boundary conditions are arbitrary positive height α and zero mean curvature.
Keywords. Natural boundary conditions, Willmore surfaces of revolution.
AMS classification. 49Q10; 53C42, 35J65, 34L30.
1
Introduction
For a smooth, immersed surface Γ ⊂ R3 and real parameters γ, µ, H0 , Nitsche in [N1] and [N2]
considered the functional
Z
F(Γ) = Φ(H, K) dA with Φ(H, K) = µ + (H − H0 )2 − γK,
(1.1)
Γ
where H is the mean curvature of the immersion, K its Gauss curvature, and dA its area element.
In many applications, Γ is an idealised model for the interface occurring in real materials. The
energy F(Γ) then reflects the surface tension and, therefore, elastic properties of this interface.
Similar versions of this functional as model for elastic energies of thin plates were already studied
by Poisson [P] in 1812, or Germain [G] in 1921. For a concise presentation we refer to Love’s
textbook [L]. In 1973, Helfrich [H] studied a functional quite similar to F from (1.1) as a model
for biological bilayer membranes, see also [O] for a more recent survey on this subject. Therefore,
F is sometimes referred to as Helfrich functional. Detailed historical information can also be found
in Nitsche [N1] and [N2].
From the mathematical point of view it is natural to assume a certain definiteness condition for
the functional F. More precisely, we require existence of a constant C > −∞ such that F(Γ) ≥ C
∗
Financial support of “Deutsche Forschungsgemeinschaft” for the project “Randwertprobleme für Willmoreflächen
- Analysis, Numerik und Numerische Analysis” (DE 611/5.1) is gratefully acknowledged
1
holds true for all connected and orientable surfaces of regularity class C 2 . As shown in [N1], this
condition imposes the following restrictions on the parameters
µ ≥ 0,
γH02 ≤ µ(1 − γ).
0 ≤ γ ≤ 1,
In the present work we study the special case H0 = µ = 0, where F takes the form
Z
Z
2
Wγ (Γ) := H dA − γ K dA, 0 ≤ γ ≤ 1.
(1.2)
Γ
Γ
This functional models the elastic energy of thin shells. Willmore in [W] studied and popularised
the functional W0 , by now called Willmore functional.
Note that for γ ∈ [0, 1], the functional Wγ is positive semidefinite. To see this, let κ1 , κ2 ∈ R
denote the principal curvatures of the surface. Then we compute
4(H 2 − γK) = (κ1 + κ2 )2 − 4γκ1 κ2 = (1 − γ)(κ1 + κ2 )2 + γ(κ1 − κ2 )2 ≥ 0
for γ ∈ [0, 1]
which gives the semi-definiteness. Moreover, strict inequality Wγ (Γ) > 0 holds for every nonplanar surface Γ if 0 < γ < 1.
We are mainly interested in minima or critical points of Wγ . Such critical points Γ ⊂ R3 have
to satisfy the Willmore equation
∆Γ H + 2H(H 2 − K) = 0
on Γ,
(1.3)
where ∆Γ denotes the Laplace-Beltrami operator on Γ, see i.e. [W]. A solution of this non-linear
fourth-order differential equation is called Willmore surface. Note that the Euler-Lagrange equation is independent of the value of γ since the integral over the Gauss curvature only contributes
to the boundary terms on account of the theorem of Gauss-Bonnet.
Existence and regularity results for closed Willmore surfaces of prescribed genus are extensively studied in the literature (see e.g., [BK], [KS1], [KS2], [LPP], [Sn] and [R]), while existence of
Willmore surfaces with prescribed boundaries is by far less studied. In the presence of boundaries
the partial differential equation (1.3) has to be accompanied by appropriate boundary conditions.
Possible choices for them are presented in [N1] and [N2] along with corresponding existence results.
Nitsche’s results are based on perturbation arguments and require certain smallness conditions on
the boundary data. On the other hand, Schätzle in [Sch] recently proved existence and regularity
of branched Willmore immersions in Sn satisfying prescribed boundary conditions. By working in
Sn some compactness problems could be overcome.
To present a complete analysis of at least special Willmore surfaces satisfying prescribed boundary conditions, we restrict ourselves to surfaces of revolution generated by rotating a symmetric
graph in the [x, y]-plane about the x-axis. Existence and classical regularity of those axially symmetric Willmore surfaces with arbitrary symmetric Dirichlet boundary conditions were recently
proved in [DDG] and [DFGS]. With the paper at hand we continue these studies, and we solve the
existence problem for Willmore surfaces of revolution with position prescribed at the boundary
and the second boundary condition they satisfy is the natural one when considering critical points
of the Willmore functional in the class of surfaces of revolution generated by symmetric graphs
where only the position at the boundary is fixed.
1.1
Main result
We consider surfaces of revolution Γ ⊂ R3 generated by rotating the graph of a smooth symmetric
function u : [−1, 1] → (0, ∞) about the x-axis. Within this class of surfaces we look for solutions
2
of the Willmore equation (1.3) under the boundary conditions
u(±1) = α > 0
and H(±1) =
γ
α
p
1 + u0 (±1)2
for γ ∈ [0, 1].
Our main result is the following.
Theorem 1.1 (Existence and regularity). For each α > 0 and for each γ ∈ [0, 1], there exists a
positive and symmetric function u ∈ C ∞ ([−1, 1], (0, ∞)), i.e. u(x) > 0 and u(x) = u(−x), such
that the corresponding surface of revolution Γ ⊂ R3 solves

2

 4Γ H + 2H(H − K) = 0 on Γ,
(1.4)
γ

.
 u(±1) = α and H(±1) = p
α 1 + u0 (±1)2
This fourth-order system along with its natural boundary conditions can be found i.e. in [N1],
[N2], or von der Mosel [vM]. In Appendix A we recall how the second boundary condition in (1.4)
arises as natural boundary condition for the functional Wγ .
For special values of α and γ,
√explicit solutions of problem (1.4) are known. For example, if γ = 1
then the circular arc u(x) = α2 + 1 − x2 provides an explicit solution of (1.4) for arbitrary α > 0.
Next, let us define some real number α∗ by
α∗ := min
y>0
with
cosh(y)
1
= ∗ cosh(b∗ ) ≈ 1.5088795 . . .
y
b
b∗ ≈ 1.1996786 . . .
solving
b∗ tanh(b∗ ) = 1.
(1.5)
(1.6)
In case γ = 0 and α > α∗ , there exist two catenoid solutions of (1.4) of the form u(x) = cosh(bx)/b,
b > 0 suitably chosen. These two solutions yield surfaces with vanishing mean curvature, i.e.
minimal surfaces. Moreover, these explicit examples show that the solutions of problem (1.4) are,
in general, not unique. Theorem 1.1 becomes particularly interesting for γ = 0 and α < α∗ , as
catenoid solutions do no longer exist under
√ this assumption. For γ = 0 and α = 1 there still exists
an explicit solution given by u(x) = 2 − 2 − x2 , a piece of the well-known Clifford torus.
So we have the following rough picture:
→ non-minimal solutions for α < α∗ ;
→ exactly one minimal surface solution for α = α∗ ;
→ two minimal surface solutions for α > α∗ ;
Existence of rotationally symmetric Willmore surfaces solution of (1.4) for γ = 0 and for all
values of α was observed numerically by Fröhlich [F] in 2004, and by Kastian [Ka] as well as
Grunau and Deckelnick [DG]. Moreover, in [Ka] the presence of a third solution for α > α∗ was
numerically observed, suggesting that α∗ is a bifurcation point on the branch of minimal surface
solutions. Recently, in [DG] Deckelnick and Grunau prove that α∗ is indeed a bifurcation point
and so, at least locally, the existence also of a non-minimal solution for α > α∗ is settled. In the
same paper, by a linearisation around the Clifford torus they prove existence of a solution to (1.4)
for γ = 0 and α near to 1. Here we extend this result by proving existence of solutions for all
α ∈ (0, α∗ ).
Also the case γ = 1 is special. Up to some constant, W1 (u) equals the total elastic energy
of u considered as a curve in the hyperbolic half-plane R2+ := {(x, y) ∈ R2 : y > 0} equipped
with the metric ds2h := y12 (dx2 + dy 2 ) (see i.e. [BG], [DDG]). Thus, varying γ within [0, 1],
we interpolate between the “Euclidean” Willmore functional with γ = 0, and the “hyperbolic”
Willmore functional for γ = 1.
3
The proof of Theorem 1.1 is based on the existence results from [DFGS] for symmetric Willmore
surfaces of revolution satisfying Dirichlet boundary conditions u(±1) = α and ∓u0 (±1) = β for
α > 0 and β ∈ R arbitrary. We construct a solution of (1.4) by minimising the Willmore energy
for fixed α and variable β. Essential tools are the continuity and the monotonicity of the Willmore
energy in β.
2
2.1
Notations. Dirichlet boundary value problem
Surfaces of revolution
We consider functions u ∈ C 4 ([−1, 1], (0, ∞)). Rotating the curve (x, u(x)) ⊂ R2 about the x-axis
generates a surface of revolution Γ ⊂ R3 which can be parametrised by
¡
¢
Γ : f (x, ϕ) = x, u(x) cos ϕ, u(x) sin ϕ ∈ R3 ,
x ∈ [−1, 1], ϕ ∈ [0, 2π).
(2.1)
The term “surface” always refers to the mapping f as well as to the set Γ. The condition u > 0
implies that f is embedded in R3 and in particular immersed.
3
Let κ1 p
and κ2 denote the principal curvatures of Γ ⊂ R3 , i.e. κ1 = −u00 (x)(1 + u0 (x)2 )− 2 and
κ2 = (u(x) 1 + u0 (x)2 )−1 . Its mean curvature H and Gaussian curvature K are
H
=
κ1 + κ2
u00 (x)
1
p
= −
,
+
3/2
0
2
2
2(1 + u (x) )
2u(x) 1 + u0 (x)2
K = κ1 κ2 = −
u00 (x)
.
u(1 + u0 (x)2 )2
For the total Gauss curvature we have
Z
Z
1
K dA = −2π
−1
Γ
¯1
¯
¯
p
dx
=
−2π
¯ ,
3
0 (x)2 ¯
0
2
2
1
+
u
(1 + u (x) )
−1
u00 (x)
u0 (x)
(2.2)
i.e. the Gauss-Bonnet theorem in our special situation. The integral is already determined by the
boundary values u0 (−1), u0 (1). This fact will become essential for the Dirichlet problem discussed
in Section 2.3. Furthermore, Wγ (Γ) takes the form
Z1 Ã
u00 (x)
1
p
Wγ (u) := Wγ (Γ) =
−
3/2
0
2
(1 + u (x) )
u(x) 1 + u0 (x)2
−1
¯1
¯
u0 (x)
¯
+2πγ p
¯ .
0
2
1 + u (x) ¯
π
2
!2
u(x)
p
1 + u0 (x)2 dx
(2.3)
−1
Remark 2.1. An important property of the energy Wγ is its rescaling invariance, i.e. given
a positive function u ∈ C 1,1 ([−r, r], (0, ∞)) for some r > 0, then the rescaled function v(x) =
u(rx)/r ∈ C 1,1 ([−1, 1], (0, ∞)) has the same energy as u, that is,
π
Wγ (v) =
2
Zr Ã
−r
u00 (x)
1
p
−
3/2
0
2
(1 + u (x) )
u(x) 1 + u0 (x)2
4
!2
¯r
¯
0 (x)
p
u
¯
u(x) 1 + u0 (x)2 dx+2πγ p
¯ .
0
2
1 + u (x) ¯−r
2.2
Notation
For α > α∗ , α∗ defined in (1.5), the following two numbers
n
o
n
o
cosh b
cosh b
b1 (α) := inf b > 0 :
≤α
and b2 (α) := sup b > 0 :
≤α
b
b
(2.4)
are well-defined and satisfy the inequality 0 < b1 (α) < b∗ < b2 (α) < +∞ with b∗ from (1.6).
Definition 2.2. For α > 0 and β ∈ R we introduce the space of functions
©
ª
N α,β := u ∈ H 2 ([−1, 1]) : u(x) > 0 , u(x) = u(−x) , u(±1) = α and u0 (−1) = β
along with
©
ª
T γ,(α,β) := inf Wγ (u) : u ∈ N α,β for γ ∈ [0, 1] .
Due to technical reasons we shall not work within N α,β , but within the smaller space
©
Nα,β := u ∈ N α,β : if α > α∗ and − α < β
ª
then u0 (x) < α in [0, 1]
(2.5)
with
©
ª
Tγ,(α,β) := inf Wγ (u) : u ∈ Nα,β for γ ∈ [0, 1] .
(2.6)
One easily sees that the space Nα,β is never empty and hence Tγ,(α,β) well-defined. That the
energy Tγ,(α,β) is attained for all α > 0, β ∈ R and γ ∈ [0, 1] is a consequence of the results in
[DFGS].
Remark 2.3. If α ≤ α∗ or −α ≥ β then the spaces Nα,β and N α,β coincide and hence Tγ,(α,β) =
T γ,(α,β) . Moreover, if α > α∗ and − sinh(b1 (α)) ≤ β then the equality T γ,(α,β) = Tγ,(α,β) again
holds (compare [DFGS]) even though the space Nα,β is now a proper subspace of N α,β . In case of
−α < β < − sinh(b1 (α)), the number Tγ,(α,β) may be strictly bigger than T γ,(α,β) .
2.3
The Dirichlet boundary value problem
In this section we recall the existence result for the Dirichlet boundary value problem (2.7) below
from [DFGS]. First, this result holds true for γ = 0 and γ = 1. At the same time a solution to this
problem is a critical point for Wγ independently of γ because, on account of (2.2), the total Gauss
curvature is a constant depending only on β. Furthermore, we state monotonicity properties of
the minimal energy Tγ,(α,β) in α.
Theorem 2.4. ([DFGS, Th.1.1]) For each α > 0 and for each β ∈ R, there exists a positive
and symmetric function u ∈ C ∞ ([−1, 1], (0, ∞)) such that the corresponding surface of revolution
Γ ⊂ R3 solves
(
4Γ H + 2H(H 2 − K) = 0 on Γ,
(2.7)
u(±1) = α,
u0 (−1) = −u0 (1) = β
and satisfies W0 (u) = T0,(α,β) . Moreover, u has the following properties:
1. If β ≥ 0, then u0 > 0 in (−1, 0).
2. If β < 0, then u0 has at most three critical points in [−1, 1].
5
In [DFGS] the monotonicity behaviour of T1,(α,β) in α for fixed β was studied. Those values of
α and β, for which a catenoid or an arc of a circle solve (2.7), mark points where the monotonicity
of this optimal energy w.r.t. α changes qualitatively. In particular, for β > 0 and α = β −1 , a
solution to (2.7) is an arc of the circle with center at the origin and going through (1, α), while for
β < 0 and α = αβ with
p
1 + β2
αβ :=
(2.8)
arsinh(−β)
the catenoid u(x) = cosh(bx)/b, b = arsinh(−β) is a minimal surface solution to (2.7). Since
β
Tγ,(α,β) = T1,(α,β) + 4π(1 − γ) p
,
1 + β2
the minimal energies Tγ,(α,β) and T1,(α,β) show the same monotonicity behaviour w.r.t. α as long
as we keep γ and β fixed. Thus, the monotonicity results from [DFGS] on T1,(α,β) yield directly
Proposition 2.5. Let γ ∈ [0, 1] be fixed.
1. For β > 0 and α > α0 ≥
1
β
it holds Tγ,(α,β) > Tγ,(α0 ,β) .
2. For β > 0 and 0 < α0 < α ≤
1
β
it holds Tγ,(α,β) < Tγ,(α0 ,β) .
3. For β = 0 and 0 < α0 < α it holds Tγ,(α,β) < Tγ,(α0 ,β) .
4. For β < 0 and 0 < α0 < α ≤ αβ it holds Tγ,(α,β) < Tγ,(α0 ,β) .
5. For β < 0 and α > α0 ≥ αβ it holds Tγ,(α,β) > Tγ,(α0 ,β) .
To prove Theorem 1.1 we make use of various important a priori estimates for solutions to (2.7)
established in [DFGS]. Let us recall them here. The real numbers b1 , b2 are defined in (2.4) and
α∗ is defined in (1.5).
Proposition 2.6. Let α > 0, β ∈ R and u ∈ C ∞ ([−1, 1], (0, ∞)) be the function from Theorem
2.4 such that the corresponding surface of revolution Γ ⊂ R3 solves (2.7). Then u has the following
qualitative properties:
1. If α ≤ α∗ then
(
)
p
1 + β2
|u (x)| ≤ max |β|, α ,
,
α
0
∗
(
p
1
α
max{|β|, α∗ } 1
(α + max{1, |β|})2 − x2 ≥ u(x) ≥ min α, p
,
, ∗
2 1 + β2
eC2 − 1
b
)
with C2 = 8(1 + max{|β|, α∗ }2 ).
2. If α > α∗ then
(
0
|u (x)| ≤ max sinh(b2 (α)), |β|,
p
1 + β2
α
)
,
(
)
∗} 1
p
1
α
sinh(b
(α))
max{|β|,
α
2
(α + max{1, |β|})2 − x2 ≥ u(x) ≥ min α, p
,
,
,
2 1 + β2
eC1 − 1
eC2 − 1
b2
with C2 = 8(1 + max{|β|, α∗ }2 ) and C1 = 2 cosh(2b2 )(1 + arsinh(|β|)(α − α∗ )).
6
(i) If − sinh(b1 (α)) ≥ β > −α, we have
0 ≤ u0 (x) ≤ −β
in [0, 1].
(ii) If β > − sinh(b1 (α)), we have
−
p
1
1
0
≤
u
(x)
≤
sinh(b
)
in
[0,
1]
and
cosh(b1 x).
α2 + 1 − x2 ≥ u(x) ≥
1
∗
α
b1
Combining the results of Theorem 2.4 and Proposition 2.6 we obtain
Corollary 2.7. Given any γ ∈ [0, 1], α > 0 and β ∈ R, there exists some function u ∈
C ∞ ([−1, 1], R) ∩ Nα,β such that the corresponding surface of revolution solves (2.7) and moreover
Wγ (u) = Tγ,(α,β) holds.
3
Continuity and monotonicity of the energy in β
In this section we analyse the behaviour of the optimal energy Tγ,(α,β) w.r.t. β if α and γ are fixed.
The results we obtain are the main ingredients for the proof of Theorem 1.1.
3.1
Continuity in β
Lemma 3.1. Let γ ∈ [0, 1] be fixed. If α ≤ α∗ , then β 7→ Tγ,(α,β) is upper semi-continuous for
β ∈ R. If α > α∗ , then β 7→ Tγ,(α,β) is upper semi-continuous for β ∈ R\{−α}.
Proof. Given u ∈ Nα,β and ε ∈ R consider the symmetric function uε (x) := u(x) + 2ε (1 − x2 )
with the properties uε (±1) = α, u0ε (−1) = β + ². Then uε ∈ N α,β+ε will hold for |ε| < ε0 ,
ε0 > 0 sufficiently small (to have uε (x) > 0 in [−1, 1]). If either α ≤ α∗ or −β > α this implies
uε ∈ Nα,β+ε (compare Remark 2.3). If, on the other hand, α > α∗ and β > −α then u ∈ Nα,β
implies u0 (x) < α in [0, 1] by Definition 2.2. This implies u0ε (x) < α in [0, 1] for |ε| ≤ ε1 and thus
uε ∈ Nα,β+² , if 0 < ε1 ≤ ε0 is chosen sufficiently small. The continuity of the mapping ε 7→ Wγ (uε )
gives
Tγ,(α,β) =
inf
u∈Nα,β
£
¤
lim Wγ (uε ) ≥
ε→0
inf
£
u∈Nα,β
¤
lim sup Tγ,(α,β+ε) = lim sup Tγ,(α,β+ε) ,
ε→0
ε→0
which just means that β 7→ Tγ,(α,β) is upper semi-continuous.
The proof of lower semi-continuity of β 7→ Tγ,(α,β) is more involved and requires the a priori
estimates from Proposition 2.6.
Lemma 3.2. Let γ ∈ [0, 1] be fixed. If α ≤ α∗ , then β 7→ Tγ,(α,β) is lower semi-continuous for
β ∈ R. If α > α∗ , then β 7→ Tγ,(α,β) is lower semi-continuous for β ∈ R\{−α}.
Proof. Because of
β
Tγ 0 ,(α,β) = Tγ,(α,β) + 4π(γ − γ 0 ) p
1 + β2
it suffices to prove the result for one particular γ, we take γ0 := 12 . Let (βk )k∈N ⊂ R be some
sequence converging to some β ∈ R. Moreover, let uk be the function from Corollary 2.7 satisfying
uk ∈ Nα,βk and Wγ0 (uk ) = Tγ0 ,(α,βk ) .
7
Since (βk )k∈N is uniformly bounded, Proposition 2.6 yields positive constants ci , i = 1, 2, 3, depending only on α such that
0 < c1 ≤ uk (x) ≤ c2 and |u0k (x)| ≤ c3
in [−1, 1]
(3.9)
holds true for all k ∈ N. If moreover α > α∗ and −α < β, then Proposition 2.6 yields additionally
©
ª
u0k (x) ≤ max − βk , sinh(b1 (α))
in [0, 1].
(3.10)
From the upper semi-continuity of Lemma 3.1 we deduce that Tγ0 ,(α,βk ) = Wγ0 (uk ) ≤ c4 holds
with some constant c4 . This is true since upper semi-continuous functions achieve a maximum on
compact sets. These estimates imply
c4 ≥ Wγ0 (uk ) =
≥
Z
1
Ã
u00k (x)2 uk (x)
1
p
5 +
0
2
uk (x) 1 + u0k (x)2
−1
(1 + uk (x) ) 2
Z 1
π
c1
u00 (x)2 dx.
2 (1 + c2 ) 25 −1 k
3
π
2
!
dx
(3.11)
Notice that due to the choice γ0 = 12 there are no boundary terms. From (3.11) we obtain uniform
boundness of the sequence in H 2 ([−1, 1]), and, after passing to a subsequence, Rellich’s embedding
theorem ensures the existence of u ∈ H 2 ([−1, 1]) such that
uk * u in H 2 ([−1, 1]) and uk → u in C 1 ([−1, 1], R).
The convergence in C 1 ([−1, 1]) ensures that u satisfies also the bounds in (3.9), in particular
u(x) > 0 in [−1, 1]. Moreover, if α > α∗ and −α < β then estimate (3.10) yields
©
ª
u0 (x) ≤ max − β, sinh(b1 (α)) < α
in [0, 1]
and hence u ∈ Nα,β . The strong convergence in C 1 ([−1, 1]) and the weak convergence in H 2 ([−1, 1])
yield
Wγ0 (uk ) =
≥
π
2
Z
π
2
Z
1
Ã
u00k (x)2 u(x)
5
(1 + u0 (x)2 ) 2
−1
1
Ã
u00 (x)2 u(x)
5
−1
(1 + u0 (x)2 ) 2
+
+
1
!
dx + o(1)
p
u(x) 1 + u0 (x)2
1
p
u(x) 1 + u0 (x)2
!
dx + o(1) = Wγ0 (u) + o(1).
Together with u ∈ Nα,β this shows
Tγ0 ,(α,β) ≤ Wγ0 (u) ≤ lim inf Wγ0 (uk ) = lim inf Tγ0 ,(α,βk )
k→∞
k→∞
proving the claimed lower semi-continuity.
The combination of the above two results now yields
Corollary 3.3. Let γ ∈ [0, 1], α > 0 be fixed. Then β 7→ Tγ,(α,β) is continuous in R if α ≤ α∗
while for α > α∗ it is continuous in R\{−α}.
8
3.2
Monotonicity results for large and small β
In this section we show that β 7→ Tγ,(α,β) is an increasing function for sufficiently large positive
values of β and a decreasing function for sufficiently small negative values of β. This allows us to
restrict to ‘bounded’ values of β when looking for the absolute minimiser.
Lemma 3.4. If γ ∈ [0, 1] and β > β 0 ≥ α−1 , then Tγ,(α,β) > Tγ,(α,β 0 ) .
Proof. By Corollary 2.7 there exists some u ∈ Nα,β such that Wγ (u) = Tγ,(α,β) . Since u0 (−1) =
β > β 0 , u0 (0) = 0, u0 is continuous and u0 ≥ 0 in [−1, 0] (see Theorem 2.4), there exists x∗ ∈ (−1, 0)
such that u0 (x∗ ) = β 0 and u(x∗ ) > α|x∗ |. We then consider the function w ∈ C 1,1 ([−1, 1]) which
is equal to u|[x∗ ,−x∗ ] rescaled to [−1, 1]. By construction, w0 (−1) = β 0 and w(±1) > α. By the
rescaling invariance of the energy (Remark 2.1) and Proposition 2.5 (note w(±1) > α ≥ β 0−1 ) we
finally get
Tγ,(α,β) = Wγ (u) ≥ Wγ (w) ≥ Tγ,(w(±1),β 0 ) > Tγ,(α,β 0 ) .
Here we have used Wγ (u) ≥ Wγ (w) which follows from γ ∈ [0, 1].
Remark 3.5. Actually, this result can be generalised to the case −∞ < γ ≤ 1. Using Corollary 2.7,
(2.3) and Lemma 3.4 we find
β
β0
Tγ,(α,β) = T1,(α,β) + 4π(1 − γ) p
> T1,(α,β 0 ) + 4π(1 − γ) p
= Tγ,(α,β 0 ) ,
1 + β2
1 + β 02
since 1 − γ ≥ 0 and β > β 0 ≥ α−1 > 0.
In the following, we set
½
0
if α < α∗ ,
β2 (α) :=
− sinh(b2 ) if α ≥ α∗ ,
with b2 = b2 (α) defined in (2.4).
(3.12)
A simple computation shows that, for β < 0, α > αβ implies β > β2 (α).
Lemma 3.6. Let γ ∈ [0, 1] be fixed. If β 0 < β ≤ min{−α, β2 (α)}, then Tγ,(α,β) < Tγ,(α,β 0 ) .
Proof. By Corollary 2.7 there exists some u ∈ Nα,β 0 such that Wγ (u) = Tγ,(α,β 0 ) . Because of
β 0 < −α there exists x ∈ (−1, 0) the smallest element such that u(x) = −αx. Since u0 (−1) = β 0 <
β ≤ −α, u0 (x) ≥ −α, and u0 is continuous, there exists x∗ ∈ (−1, x) such that u0 (x∗ ) = β and
u(x∗ ) < α|x∗ |. We consider the function w ∈ C 1,1 ([−1, 1]) equal to u|[x∗ ,−x∗ ] rescaled to [−1, 1].
By construction there hold w0 (−1) = β and w(±1) < α. If α > α∗ the assumption β ≤ β2 (α) gives
α ≤ αβ with αβ defined in (2.8). While if α ≤ α∗ then clearly also α ≤ αβ . In both cases, Remark
2.1 and Proposition 2.5 (noting that w(±1) < α ≤ αβ ) yield
Tγ,(α,β 0 ) = Wγ (u) ≥ Wγ (w) ≥ Tγ,(w(±1),β) > Tγ,(α,β) .
Here we have used Wγ (u) ≥ Wγ (w) due to γ ∈ [0, 1].
Remark 3.7. This result still holds for all γ ≥ 0 because
β0
β
Tγ,(α,β 0 ) = T0,(α,β 0 ) − 4πγ p
> T0,(α,β) − 4πγ p
= Tγ,(α,β) ,
1 + β 02
1 + β2
which holds for γ ≥ 0 and β 0 < β ≤ min{−α, β2 (α)}.
9
3.3
The case γ = 0
For this case we give a complete description of the monotonicity behaviour of β 7→ T0,(α,β) for all
values of β. For α ≤ α∗ this mapping is decreasing on (−∞, −α] while it is increasing on [−α, ∞).
For α > α∗ the behaviour is more complicated due to the presence of the two catenoid solutions
whose energy W0 is zero.
Similarly to β2 (α), let us introduce
½
−α∗
if α < α∗ ,
β1 (α) :=
with b1 = b1 (α) defined in (2.4).
(3.13)
− sinh(b1 ) if α ≥ α∗ ,
Lemma 3.8. If α−1 ≥ β > β 0 ≥ max{−α, β1 (α)}, then T0,(α,β) > T0,(α,β 0 ) .
Proof. Given β > β 0 we first define b := −arsinh(β), b0 := −arsinh(β 0 ) and note b < b0 . By
Corollary 2.7 there exists some u ∈ Nα,β such that W0 (u) = T0,(α,β) . Let f (x) be the catenary
with initial data f (−1) = α, f 0 (−1) = β, i.e. the function
µ
¶
cosh b
α
cosh
(x + 1) − b .
(3.14)
f (x) =
cosh b
α
At the point
α
(b − b0 )
(3.15)
cosh b
we have f 0 (x∗ ) = β 0 . Note that x∗ < −1 since b < b0 . Let v ∈ C 1,1 ([x∗ , −x∗ ]) be the symmetric
function equal to f on [x∗ , −1] and equal to u in (−1, 0]. Furthermore, let w ∈ C 1,1 ([−1, 1]) be
v rescaled to [−1, 1]. By construction it holds w0 (−1) = β 0 . In order to apply the monotonicity
property of the energy in α we need to show w(±1) < α. Since β > β 0 ≥ −α we have sinh(x) < α
for all x ∈ (b, b0 ), and hence
x∗ := −1 +
cosh(b0 ) − cosh(b)
<α
b0 − b
or equivalently
cosh(b0 )
< 1.
cosh(b) − α(b − b0 )
From this inequality we see
w(±1) =
v(x∗ )
α cosh(b0 )
=
|x∗ |
cosh(b) 1 −
1
α
cosh(b) (b
− b0 )
=α
cosh(b0 )
< α.
cosh(b) − α(b − b0 )
(3.16)
Since the piece of catenoid has zero energy for γ = 0, and the energy Wγ is invariant under rescaling
(Remark 2.1), we obtain
T0,(α,β) = W0 (u) = W0 (w) ≥ T0,(w(±1),w0 (−1)) = T0,(w(−1),β 0 ) .
Now, if β 0 ≥ 0, then, using w(±1) < α < β 0−1 , Proposition 2.5 yields T0,(w(±1),β 0 ) > T0,(α,β 0 ) . On
the other hand, if β 0 < 0 we claim α ≤ αβ 0 . This is clear if α ≤ α∗ , while if α > α∗ it is assured
by the assumption β 0 ≥ − sinh(b1 ) which follows from the definition of β1 (α). Proposition 2.5 now
yields again T0,(w(±1),β 0 ) > T0,(α,β 0 ) . In both cases we obtain T0,(α,β) > T0,(α,β 0 ) .
Combining our Lemmata 3.4, 3.6 and 3.8 with Corollary 3.3 for α ≤ α∗ we obtain:
Corollary 3.9. If 0 < α ≤ α∗ , then T0,(α,β) is increasing in β for β ∈ [−α, ∞) and decreasing for
β ∈ (−∞, −α]. The mapping β 7→ T0,(α,β) achieves its global minimum at β = −α.
We still have to discuss the case α > α∗ and β ∈ [β2 (α), β1 (α)]. Here, the monotonicity
behaviour becomes more involved due to the presence of the two catenoids corresponding to the
two values β1 (α) and β2 (α) for the boundary datum β.
10
Lemma 3.10. If α > α∗ and −α ≥ β > β 0 ≥ β2 (α), then T0,(α,β) > T0,(α,β 0 ) .
Proof. Note that the assumption −α > β 0 ≥ β2 (α) yields α ≥ αβ 0 . The claim is proven quite
similarly as Lemma 3.8. By Corollary 2.7 there exists some u ∈ Nα,β such that W0 (u) = T0,(α,β) .
Let f (x) be the function from (3.14). At x∗ defined as in (3.15), we have f 0 (x∗ ) = β 0 . Now
consider the symmetric function v ∈ C 1,1 ([x∗ , −x∗ ]) which is equal to f in [x∗ , −1] and to u in
(−1, 0]. Furthermore, let w ∈ C 1,1 ([−1, 1]) be its rescaling to [−1, 1]. By construction, w0 (−1) = β 0
and w(±1) > α. Indeed, since −α ≥ β > β 0 we have sinh(x) > α for all x ∈ (b, b0 ) and also
cosh(b0 ) − cosh(b)
>α
b0 − b
or equivalently
cosh(b0 )
> 1.
cosh(b) − α(b − b0 )
This inequality implies w(±1) > α which is proven just like (3.16). We then obtain the inequality
T0,(α,β) = W0 (u) = W0 (w) ≥ T0,(w(±1),β 0 ) > T0,(α,β 0 ) ,
using Proposition 2.5 together with w(±1) > α > αβ 0 .
For α > α∗ and β ∈ (−α, β1 (α)] the elements u ∈ Nα,β have the additional restriction u0 (x) < α
in [0, 1] (compare Definition 2.2). This property shall be used now in order to prove monotonicity
in β ∈ (−α, β1 (α)].
Lemma 3.11. If α > α∗ and β1 (α) ≥ β > β 0 > −α, then T0,(α,β) < T0,(α,β 0 ) .
Proof. Note that the assumption −α < β ≤ β1 (α) yields α ≥ αβ . By Corollary 2.7 there exists
some u ∈ Nα,β 0 such that W0 (u) = T0,(α,β 0 ) . Moreover, β 0 ≤ u0 (x) ≤ 0 in [−1, 0] by Proposition 2.6.
From this estimate together with β 0 > −α it follows that u(x) > α|x| for x ∈ (−1, 1). Since
u0 (−1) = β 0 < β < 0, u0 (0) = 0 and u0 is continuous, there exists x∗ ∈ (−1, 0) such that u0 (x∗ ) = β
and u(x∗ ) > α|x∗ |. We consider then the function w ∈ C 1,1 ([−1, 1]) that is equal to u|[x∗ ,−x∗ ]
rescaled to [−1, 1]. By construction, w0 (−1) = β and w(±1) > α. Proposition 2.5 yields
T0,(α,β 0 ) = W0 (u) ≥ W0 (w) ≥ T0,(w(±1),β) > T0,(α,β) .
Combining the previous results we have:
Corollary 3.12. For fixed α > α∗ , the function β 7→ T0,(α,β) is decreasing on the intervals
(−∞, β2 (α)] and (−α, β1 (α)] while it is increasing on the intervals [β2 (α), −α] and [β1 (α), +∞).
3.4
The case γ ∈ [0, 1]
A combination of Lemmata 3.4 and 3.6 yields:
Corollary 3.13. For γ ∈ [0, 1] and α ≤ α∗ the mapping β 7→ Tγ,(α,β) is decreasing for −∞ < β ≤
−α and increasing for α−1 ≤ β < +∞.
This result does not give us any information about the monotonicity if −α < β < α−1 . We
may expect that there exists a unique β̃ = β̃(α, γ) ∈ [−α, α−1 ] such that β 7→ Tγ,(α,β) is decreasing
on (−∞, β̃] and increasing on [β̃, +∞). In fact, this claim is true for γ = 0 with β̃ = −α, due to
Corollary 3.9. It is also true for γ = 1 where we can take β̃ = α−1 .
Corollary 3.14. For γ ∈ [0, 1] and α > α∗ the mapping β 7→ Tγ,(α,β) is decreasing on (−∞, β2 (α)]
and (−α, β1 (α)], and increasing on [α−1 , ∞).
11
Proof. For β ≥ α−1 and β ≤ β2 (α) the claim follows from Lemmata 3.4 and 3.6. For β ∈
(−α, β1 (α)], we note
β
Tγ,(α,β) = T0,(α,β) − 4πγ p
.
1 + β2
√
Since T0,(α,β) is decreasing by Corollary 3.12 on (−α, β1 (α)] and x 7→ −x/ 1 + x2 is also decreasing
for x < 0, for β1 (α) ≥ β > β 0 > −α we obtain
β0
β
Tγ,(α,β 0 ) = T0,(α,β 0 ) − 4πγ p
> T0,(α,β) − 4πγ p
= Tγ,(α,β) .
1 + β 02
1 + β2
The claim follows.
Similarly to Corollary 3.14, this result does not yield information on the monotonicity if β1 (α) <
β < α−1 . One may conjecture that there exists some β̃ = β̃(α, γ) ∈ [β1 (α), α−1 ] such that
β 7→ Tγ,(α,β) is decreasing on (−α, β̃] and increasing on [β̃, +∞).
Remark 3.15. Similarly to the case γ = 0 treated in in Section 3.3, we can completely discuss
the monotonicity behaviour of β 7→ T1,(α,β) for γ = 1. It is decreasing on (−∞, α−1 ] and increasing
√
on [α−1 , +∞). The minimum β = α−1 corresponds to the circular arc u(x) = α2 + 1 − x2 ,
u ∈ Nα,α−1 which has zero energy W1 (u) = 0.
The proof is quite similar to the case γ = 0. Instead of adding a piece of a catenoid, as done
in the proof of Lemmata 3.8 and 3.10, we now add a circular arc. While for γ = 0 adding a piece
of catenoid does not change the energy W0 , in the case of γ = 1 adding a circular arc does not
change the energy W1 . We point out that the procedure of adding ‘pieces’ with zero energy cannot
be used for 0 < γ < 1 since for this range of γ the energy Wγ is always larger than zero.
4
Proof of Theorem 1.1
We first study the case α ≤ α∗ . Setting β − := min{−α, β2 (α)}, β + := α−1 , Corollary 3.13 implies
Tγ,α := inf Tγ,(α,β) =
β∈R
inf
β − ≤β≤β +
Tγ,(α,β) .
The continuity of the energy in β, proven in Corollary 3.3, yields some β ∗ ∈ [β − , β + ] such that
Tγ,α = Tγ,(α,β ∗ ) . By Corollary 2.7 there exists some u ∈ Nα,β ∗ ∩ C ∞ ([−1, 1], (0, ∞)) such that
Wγ (u) = Tγ,(α,β ∗ ) . Since u minimises the energy Wγ within the class ∪β∈R Nα,β , it solves boundary
value problem (1.4).
Let us now study the case α > α∗ . Here we set β − := β1 (α) > −α, β + := α−1 . Corollary 3.14
yields
Tγ,α :=
inf
Tγ,(α,β) =
inf
Tγ,(α,β) .
β − ≤β≤β +
−α<β<+∞
β∗
[β − , β + ]
Again, Corollary 3.3 gives some
∈
such that Tγ,α = Tγ,(α,β ∗ ) . Corollary 2.7 yields some
∞
u ∈ Nα,β ∗ ∩ C ([−1, 1], R) such that Wγ (u) = Tγ,(α,β ∗ ) . This function u minimises the energy Wγ
within the class ∪−α<β<+∞ Nα,β and hence is solution of the boundary value problem (1.4). Here
it is crucial that β ∗ > −α.
Remark 4.1. In the particular case α ≤ α∗ and γ = 0 the monotonicity property of the energy
in β (Corollary 3.9) yields that the constructed solution of (1.4) satisfies u0 (−1) = −α. One can
verify this for the values of α for which an explicit solution to (1.4) is known. For α = 1 this
solution
√ is a piece of the Clifford torus, i.e. 0 the surface of revolution corresponding to f (x) :=
2 − 2 − x2 . One sees that f (−1) = 1 and f (−1) = −1. Another explicit solution is the catenoid
x 7→ g(x) := cosh(b∗ x)/b∗ with b∗ defined in (1.6). This function has boundary value g(−1) = α∗
with α∗ defined in (1.5) and g 0 (−1) = − sinh(b∗ ) = −α∗ by definition of b∗ and α∗ .
12
A
Natural boundary conditions
The following lemma yields the first variation of the functional Wγ .
Lemma A.1. Let u ∈ C 4 ([−1, 1], (0, ∞)). Then for all ϕ ∈ H 2 ([−1, 1]) ∩ H01 ([−1, 1]), we have
¯
·
¸
Z
¯
u(x)ϕ0 (x) 1
d
H(u + tϕ)2 dA[u + tϕ]¯¯
= −2π H(x)
dt Γ
1 + u0 (x)2 −1
t=0
Z
1
¡
¢
u(x)ϕ(x) ∆Γ H(x) + 2H(H 2 − K) dx,
−2π
−1
and
d
dt
Z
Γ
"
¯
¯
K[u + tϕ] dA[u + tϕ] ¯¯
= −2π
t=0
ϕ0 (x)
#1
(1 + u0 (x)2 )3/2
,
−1
Γ being the surface of revolution generated by u + tϕ.
The first statement was proved in [DG, Lemma 6]. The second identity follows directly if we
write the Gauss curvature K in coordinates. Thus, the first variation of the composed functional
Wγ (u) can be written as
"Ã
!
#1
¯
d
γ
u(x)ϕ0 (x)
¯
p
H(x) −
Wγ (u + tϕ) ¯
= − 2π
dt
t=0
u(x) 1 + u0 (x)2 1 + u0 (x)2 −1
Z
1
¡
¢
uϕ ∆Γ H + 2H 3 − 2HK dx
− 2π
−1
for all ϕ ∈ H 2 ([−1, 1]) ∩ H01 ([−1, 1]). The corresponding boundary value problem is given by (1.4).
In order to provide a geometric interpretation of this boundary value problem, we observe
Lemma A.2. Let x ∈ (−1, 1) be fixed, and consider the curve ϕ 7→ X(x, ϕ) on Γ. Then it holds
κn (x) =
1
p
u(x) 1 + u0 (x)2
for its normal curvature w.r.t. the surface unit normal vector
¡
¢
1
ν(x, ϕ) = u0 (x), − cos ϕ, − sin ϕ p
.
1 + u0 (x)2
Proof. Parametrising the curve by arclength s ∈ [0, 2πu(x)] gives
µ
¶
s
s
X(s) =
x, u(x) cos
, u(x) sin
,
u(x)
u(x)
µ
¶
s
s
0
, cos
,
X (s) =
0, − sin
u(x)
u(x)
µ
¶
1
s
1
s
00
X (s) =
0, −
cos
,−
sin
.
u(x)
u(x) u(x)
u(x)
Thus, the normal curvature w.r.t. ν is given by
¡
s ¢
1
1
i= p
κn (x) = hX 00 (s), ν x,
.
u(x)
1 + u0 (x)2 u(x)
13
On account of this lemma, the natural boundary data can be expressed in terms of the geometric
quantity κn at the boundary of the surface: We can write (1.4) in the form
(
∆Γ H + 2H 3 − 2HK = 0 on Γ,
u(±1) = α,
H(±1) = γκn (±1).
A detailed computation of natural boundary conditions even for Helfrich’s functional can be found
in the literature. We want to mention i.e. Nitsche [N1], [N2], and von der Mosel [vM].
B
Estimates
Lemma B.1. Let α > 0 and β ∈ R. Then the solution u ∈ C ∞ ([−1, 1], (0, ∞)) of (2.7), constructed
in the proof of Theorem 2.4, satisfies
p
u(x) ≤ (α + max{1, |β|})2 − x2 , x ∈ [−1, 1].
Proof. If αβ ≤ 1, the function u satisfies x + u(x)u0 (x) ≥ 0 in [0, 1] (see [DFGS, Lemma 3.16] for
β ≥ 0, and with a similar reasoning for β < 0). Thus,
p
p
u(x) ≤ 1 + α2 − x2 ≤ (1 + α)2 − x2 in [−1, 1].
If αβ > 1, u satisfies x + u(x)u0 (x) ≤ 0 in [0, 1] (see [DFGS, Lemma 3.9]), and integrating from 0
to x ∈ (0, 1] gives
p
u(x) ≤ u(0)2 − x2 with u(0) ≤ α + β
where the last inequality follows from |u0 (x)| ≤ β for all x ∈ [−1, 1] ([DFGS, Theorem 3.11]).
Lemma B.2. Let α > 0 and β ∈ R. Let u ∈ C ∞ ([−1, 1], (0, ∞)) denote the solution of (2.7),
constructed in the proof of Theorem 2.4. Then u restricted on [0, 1] has the following properties:
1. If β ≥ 0, then − max{β, α−1 } ≤ u0 (x) ≤ 0.
2. If β < 0, −β < α and α ≥ αβ , then 0 ≤ u0 (x) ≤ −β.
3. If β < 0, −β ≥ α and α ≥ αβ , then 0 ≤ u0 (x) ≤ sinh(b2 ) with b2 = b2 (α) from (2.4).
4. If β < 0 and α < αβ , then
p
1 + β2
−
≤ u0 (x) ≤ max {−β, α∗ } .
α
In the particular case αβ > α ≥ α∗ and β > − sinh(b1 (α)), we have
sinh(b1 x) ≥ u0 (x) ≥ −(α∗ )−1
with b1 = b1 (α) as defined in (2.4).
Proof.
1. Let β ≥ 0. If αβ ≤ 1, u satisfies x + u(x)u0 (x) ≥ 0 and u0 (x) ≤ 0 in [0, 1] ([DFGS,
Lemma 3.16]). It follows |u0 (x)| ≤ α−1 . If αβ > 1, u satisfies u0 (x) ≤ 0 and |u0 (x)| ≤ β for
all x ∈ [−1, 1] ([DFGS, Theorem 3.11]).
2. See Theorem 4.24 from [DFGS].
3. See Theorem 4.17 from [DFGS].
14
4. Following [DFGS, Lemma 4.27], u hast at most three critical points in [−1, 1], and it holds
u0 (x) ≤ max{−β, α∗ } ([DFGS, Th. 4.39] if α ≥ α∗ , [DFGS, Th. 4.48] if α < α∗ ). If u has
exactly one critical point in [−1, 1], then u0 ≥ 0 in [0, 1]. Otherwise, there is x0 ∈ (0, 1) so
that u0 (x0 ) = 0, u0 > 0 in (x0 , 1], u0 < 0 in (0, x0 ). Since x + u(x)u0 (x) ≥ 0 in [0, 1] and
u(x0 ) ≥
α
x0
arsinh(−β)(αβ − α)
(2.17)
by Lemma 4.29 from [DFGS], we get by definition of αβ
arsinh(−β)(αβ − α)
cosh(arsinh(−β))
u (x) ≥ −
≥−
=−
α
α
0
p
1 + β2
.
α
In the special case αβ > α ≥ α∗ and β > − sinh(b1 (α)), Lemma 4.36 from [DFGS] gives
u(x) ≥
1
cosh(b1 x) and u0 (x) ≤ sinh(b1 x).
b1
The estimate follows using x + u(x)u0 (x) ≥ 0 in [0, 1].
For α > α∗ we defined b1 = b1 (α) < b∗ < b2 = b2 (α) via α = cosh(bi )/bi for i = 1, 2.
Lemma B.3. Let α > 0 and β ∈ R. Let u ∈ C ∞ ([−1, 1], (0, ∞)) be the solution of (2.7) constructed in Theorem 2.4. Then u restricted on [0, 1] has the following properties:
1. If β ≥ 0, then u(x) ≥ α.
2. If β < 0 and α > αβ , then
min u(x) ≥
x∈[0,1]
sinh(b2 )
with C1 = 2 cosh(2b2 )(1 + arsinh(−β)(α − α∗ )).
eC1 − 1
(2.18)
3. If β < 0 and α = αβ , then u(x) ≥ b−1
2 .
4. If β < 0 and α < αβ , then
(
)
1
α
max{−β, α∗ }
p
u(x) ≥ min
,
with C2 = 8(1 + max{−β, α∗ }2 ).
2 1 + β2
eC2 − 1
In the particular case αβ > α ≥ α∗ and β > − sinh(b1 (α)), this estimate can be improved
with u(x) ≥ b−1
1 .
Proof.
1. By Lemma B.2 we know that if β ≥ 0, then u0 ≥ 0 and hence u(x) ≥ α.
For the other cases we recall Lemma 4.9 from [DFGS]: Let ν := maxx∈[0,1] {u0 (x)} and x0 ≥ 0
so that u0 (x0 ) = 0 and u0 > 0 in (x0 , 1]. Then there hold
Ã
!
1 p
4β
1 − x0
with C = ν 1 + ν 2 W1 (u) + p
.
(2.19)
min u(x) = u(x0 ) ≥ ν C
e −1
2
x∈[0,1]
1 + β2
2. If β < 0 and α > αβ , we have x0 = 0 and u0 (x) ≤ sinh(b2 (α)) for x ∈ [0, 1] by Lemma B.2.
Here, use that −β < sinh(b2 (α)) if −β < α. The estimate (2.18) follows from (2.19) using
the energy estimate from ([DFGS, Prop.6.8]):
−8β
W1 (u) ≤ p
(1 + arsinh(−β)(α − αβ )) ≤ 8(1 + arsinh(−β)(α − α∗ )).
1 + β2
15
3. If β < 0 and α = αβ , the solution is u(x) = cosh(b1 x)/b1 or u(x) = cosh(b2 x)/b2 with b1 ≤ b2
and hence, the estimate follows directly.
4. If β < 0 and α < αβ , we distinguish between x0 ≤ 1/2 and x0 > 1/2. In the first case,
we proceed as we did for α > αβ . By the estimate on u0 from Lemma B.2, and taking the
following energy estimate into account (see [DFGS, Prop.6.10])
µ
¶
α − αβ
−8β
W1 (u) ≤ p
+ 8 tanh arsinh(−β)
≤ 16,
α
1 + β2
inequality (2.19) gives us
min u(x) ≥
x∈[0,1]
max{−β, α∗ }
with C2 = 8(1 + max{−β, α∗ }2 ).
eC2 − 1
But if x0 ≥ 1/2, it follows from Lemma 4.29 in [DFGS] (see (2.17)) that
u(x) ≥ u(x0 ) ≥
α
α
α
1
1
1
.
≥
= p
2 arsinh(−β)(αβ − α)
2 cosh(arsinh(−β))
2 1 + β2
In the special case αβ > α ≥ α∗ and β > − sinh(b1 (α)), Lemma 4.36 from [DFGS] gives us
u(x) ≥ cosh(b1 x)/b1 , and therefore u(x) ≥ 1/b1 .
Acknowledgment. The second author thanks Klaus Deckelnick and Hans-Christoph Grunau for
useful discussions.
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Matthias Bergner, Institut für Differentialgeometrie, Gottfried Wilhelm Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany,
E-mail: [email protected]
Anna Dall’Acqua, Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, D39016 Magdeburg, Germany
E-mail: [email protected]
Steffen Fröhlich, Institut für Mathematik, Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany
E-mail: [email protected]
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